in ${ℝ}^2$

$ℒ=\left(\begin{array}{cc}15& -8\\ 24& -13\end{array}\right)\begin{array}{ccc}p\left(x\right)& =& det(x{ℐ}_2-ℒ)\\ & =& -3-2x+x^2\\ & =& (x-3)(x+1)\end{array}$

Eigenvalue ${\lambda }_1=3$

$\left(\begin{array}{ccc}-12& 8& 0\\ -24& 16& 0\end{array}\right)\stackrel{}{\to }\left(\begin{array}{ccc}3& -2& 0\\ 0& 0& 0\end{array}\right)$

Eigenvalue ${\lambda }_2=-1$

$\left(\begin{array}{ccc}-16& 8& 0\\ -24& 12& 0\end{array}\right)\stackrel{}{\to }\left(\begin{array}{ccc}2& -1& 0\\ 0& 0& 0\end{array}\right)$